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EXTREMAL HIGHER CODIMENSION CYCLES ON MODULI SPACES

OF CURVES

DAWEI CHEN AND IZZET COSKUN

Abstract. We show that certain geometrically defined higher codimension cycles are ex-tremal in the effective cone of the moduli spaceMg,n of stable genus g curves with n orderedmarked points. In particular, we prove that codimension two boundary strata are extremaland exhibit extremal boundary strata of higher codimension. We also show that the locusof hyperelliptic curves with a marked Weierstrass point in M3,1 and the locus of hyperel-liptic curves in M4 are extremal cycles. In addition, we exhibit infinitely many extremalcodimension two cycles in M1,n for n ≥ 5 and in M2,n for n ≥ 2.

Contents

1. Introduction 1

2. Preliminaries on effective cycles 3

3. Preliminaries on moduli spaces of curves 7

4. The effective cones of M3 and M3,1 9

5. The effective cones of Mg for g ≥ 4 15

6. The effective cones of M0,n 21



References 25

1. Introduction

LetMg,n denote the Deligne-Mumford-Knudsen moduli space of stable genus g curves withn ordered marked points. In this paper, we study the effective cones of higher codimensioncycles on Mg,n. We work over the field of complex numbers C.

Motivated by the problem of determining the Kodaira dimension of Mg,n, the cone of

effective divisors of Mg,n has been studied extensively, see e.g. [HMu, Harr, EH, Far, Lo,

V, CT, CC2]. In contrast, little is known about higher codimension cycles on Mg,n, in partbecause their positivity properties are not as well-behaved. For instance, higher codimensionnef cycles may fail to be pseudoeffective [DELV]. Furthermore, unlike the case of divisors,we lack simple numerical, cohomological and geometric conditions for determining whether

Date: April 28, 2015.2010 Mathematics Subject Classification. 14H10, 14C25, 14E30.During the preparation of this article the first author was partially supported by the NSF grant DMS-1200329,the NSF CAREER grant DMS-1350396, and the second author was partially supported by the NSF CAREERgrant DMS-0950951535.

1

2 DAWEI CHEN AND IZZET COSKUN

higher codimension cycles are nef or pseudoeffective. Nevertheless, there has been growinginterest in understanding the structure of effective cones of higher codimension cycles, seee.g. [FL1] for recent progress and the current state of the art in this field.

In this paper, we study the effective cone of higher codimension cycles in Mg,n. In par-ticular, we show that certain geometrically defined higher codimension cycles span extremalrays of the effective cone of Mg,n.

Main Results.

(i) Every codimension two boundary stratum of Mg and of M0,n is extremal (Theo-rems 4.3, 5.3 and 6.1).

(ii) The higher codimension boundary strata associated to certain dual graphs describedin §5 are extremal in Mg (Theorem 5.6).

(iii) Every codimension k boundary stratum ofM0,n parameterizing curves with k markedtails attached to an unmarked P1 is extremal (Theorem 6.2).

(iv) There exist infinitely many extremal effective codimension two cycles in M1,n for

every n ≥ 5 (Theorem 7.2) and in M2,n for every n ≥ 2 (Theorem 8.2).(v) The locus of hyperelliptic curves with a marked Weierstrass point is a non-boundary

extremal codimension two cycle in M3,1 (Theorem 4.6).(vi) The locus of hyperelliptic curves is a non-boundary extremal codimension two cycle

in M4 (Theorem 5.9).

These results illustrate that the effective cone of higher codimension cycles on Mg,n canbe very complicated even for small values of g and n.

In order to verify the extremality of a codimension k cycle, we use two criteria. First, wefind a criterion that shows the extremality of loci that drop the largest possible dimensionunder a morphism (Proposition 2.2). We then apply the criterion to morphisms from Mg,n

to different modular compactifications of Mg,n. Second, we use induction on dimension. To

prove that a cycle Z is extremal, we first show that Z is extremal in a divisor D ⊂ Mg,n

containing Z, and then show that effective cycles representing Z must be contained in D(Proposition 2.5). In certain cases we can strengthen the result by showing the extremalityof Z in the pseudoeffective cone (Remark 2.7). We point out at various places when it applies.

Most of the extremal cycles we consider are contained in the boundary ofMg,n. However,in Theorem 4.6 and Theorem 5.9, we show the extremality of some non-boundary cycles. Theproofs of these theorems are more delicate, relying on a detailed analysis of canonical curvesin genus three and four.

The paper is organized as follows. In Section 2 and Section 3 we review the basic propertiesof effective cycles and moduli spaces of curves, respectively. Then we carry out the study ofthe effective cones of Mg,n according to the values of g and n: g = 3 and n ≤ 1 (Section 4);g ≥ 4 and n = 0 (Section 5); g = 0 and arbitrary n (Section 6); g = 1 and n ≥ 5 (Section 7);g = 2 and n ≥ 2 (Section 8). Sections 4–8 are basically independent, so the reader may readthem in any order.

Acknowledgments. We would like to thank Maksym Fedorchuk, Mihai Fulger, Joe Harris,Brian Lehmann, John Lesieutre, Anand Patel, Luca Schaffler and Nicola Tarasca for helpfuldiscussions related to this paper. We are also grateful to the anonymous referees for providingmany useful comments and suggestions.

EXTREMAL HIGHER CODIMENSION CYCLES ON MODULI SPACES OF CURVES 3

2. Preliminaries on effective cycles

In this section, we review basic properties of the effective cone of cycles on an algebraicvariety and develop some criteria for proving the extremality of an effective cycle. Throughoutthe paper, all varieties are defined over C and all linear combinations of cycles are with R-coefficients.

Let X be a complete variety. A cycle on X is a formal sum of closed subvarieties of X.A cycle is k-dimensional if all subvarieties in the sum are k-dimensional. A cycle is effectiveif all coefficients in the sum are nonnegative. Two k-dimensional cycles A and B on X arenumerically equivalent, if A ∩ P = B ∩ P under the degree map for all weight k polynomialsP in Chern classes of vector bundles on X, where ∩ is the cap product, see [Fu, Chapter 19].When X is nonsingular, this is equivalent to requiring A · C = B · C for all subvarieties Cof codimension k, where · is the intersection product. The focus of the paper is the modulispace of curves, which is Q-factorial but may have finite quotient singularities. Nevertheless,the intersection product is still compatible with the cup product, see [E, Section 1].

Denote by [Z] the numerical class of a cycle Z. Let Nk(X) (resp. Nk(X)) denote the R-vector space of cycles of dimension k (resp. codimension k) modulo numerical equivalence. It

is a finite dimensional vector space. Let Effk(X) ⊂ Nk(X) (resp. Effk(X) ⊂ Nk(X)) denotethe effective cone of dimension k (resp. codimension k) cycles generated by all effective cycle

classes. Their closures Effk(X) and Effk(X) are called the pseudoeffective cones.

A closed convex cone is determined by its extremal rays; a general convex cone can bebetter understood by specifying its extremal rays. Recall that a ray R is called extremal, iffor every D ∈ R and D = D1 + D2 with D1, D2 in the cone, we have D1, D2 ∈ R. If anextremal ray is spanned by the class of an effective cycle D, we say that D is an extremaleffective cycle.

There is a well-developed theory to study the cone Eff1(X), including numerical, cohomo-

logical, analytic and geometric conditions for checking whether a divisor is in Eff1(X), see

[La]. In contrast, Effk(X) for k ≥ 2 is not well-understood. We first describe two simple

criteria for checking extremality of cycles in Effk(X).

Let f : X → Y be a morphism between two complete varieties. To a subvariety Z ⊂ X ofdimension k we associate an index

ef (Z) = dimZ − dim f(Z).

Note that ef (Z) > 0 if and only if Z drops dimension under f .

Proposition 2.1. Let f : X → Y be a morphism between two projective varieties and letk > m ≥ 0 be two integers. Let Z be a k-dimensional subvariety of X such that ef (Z) ≥ k−m.If [Z] = a1[Z1] + · · · + ar[Zr] ∈ Nk(X), where Zi is a k-dimensional subvariety of X andai > 0 for all i, then ef (Zi) ≥ k −m for every 1 ≤ i ≤ r.

Proof. Let A and B be two very ample divisor classes on X and on Y , respectively. ThenN = f∗B is base-point-free on X. In particular, if U ⊂ X is an effective cycle of dimensionk, then the intersection N j · [U ] is either zero or can be represented by an effective cycle ofdimension k − j for 1 ≤ j ≤ k. In the latter case, by the projection formula and the veryampleness of A, we have 0 < Ak−j · N j · [U ] = Bj · [f∗(W )], where W is an effective cyclerepresenting Ak−j · [U ]. Since ef (Z) ≥ k −m, we conclude that Nm+1 · [Z] = 0. Therefore,Nm+1 · [Zi] = 0 for 1 ≤ i ≤ r. Otherwise, Nm+1 · [Zi] can be represented by an effective cycle.


Intersecting both sides of the equality Nm+1 · [Z] = Nm+1 · (∑r

i=1 ai[Zi]) with Ak−m−1 wewould obtain a contradiction. By the projection formula, we conclude that Bm+1·(f∗[Zi]) = 0,hence dim f(Zi) ≤ m as desired. �

As a corollary of Proposition 2.1, we obtain the following extremality criterion.

Proposition 2.2. Let f : X → Y be a morphism between two projective varieties. Fix twointegers k > m ≥ 0. Among all k-dimensional subvarieties Z of X, assume that only finitelymany of them, denoted by Z1, . . . , Zn, satisfy ef (Z) ≥ k−m. If the classes of Z1, . . . , Zn arelinearly independent, then each Zi is an extremal effective cycle in Effk(X).

Proof. Suppose that [Zi] = a1[D1] + · · · + ar[Dr] with Dj a k-dimensional subvariety of Xand aj > 0 for all j. Then, by Proposition 2.1, ef (Dj) ≥ k −m. Since Z1, . . . , Zn are theonly k-dimensional subvarieties of X with index ef ≥ k −m, we conclude that Dj has to beone of Z1, . . . , Zn. Since their classes are independent, we conclude that [Dj ] is proportionalto [Zi] for all j. Therefore, Zi is extremal in Effk(X). �

We will apply Propositions 2.1 and 2.2 to morphisms from Mg,n to alternate modularcompactifications of Mg,n. We also remark that the classes of exceptional loci in highercodimension may fail to be linearly independent.

Remark 2.3. The following example, which is related to Hironaka’s example of a completebut non-projective variety (see [Hart, Appendix B, Example 3.4.1]), was pointed out to theauthors by John Lesieutre. Let X be a smooth threefold, and let C,D ⊂ X be two smoothcurves meeting transversally at two points p and q. Let Y be the blowup of X along C. Thefibers Fr of the exceptional locus over C have the same numerical class for all r ∈ C. Next,let Z be the blowup of Y along the proper transform of D. The proper transforms Ep ofFp and Eq of Fq have classes different from (the transform of) Fr for r 6= p, q. However, Epand Eq have the same class, and their normal bundles are isomorphic to O(−1) ⊕ O(−1).Hence, they are flopping curves if we blow up D first and then blow up the proper transformof C. In particular, there exists a small contraction Z → W such that the exceptional locusconsists of Ep and Eq only.

The next corollary will be useful when studying the effective cones of the boundary strata.

Corollary 2.4. Let X and Y be projective varieties such that numerical equivalence andrational equivalence are the same for codimension k cycles in X, Y and X × Y , respectively,with R-coefficients. Suppose Z is an extremal effective cycle of codimension k in X. ThenZ × Y is an extremal effective cycle of codimension k in X × Y .

Proof. Suppose [Z×Y ] =∑ai[Ui], where ai > 0 and Ui are irreducible subvarieties of X×Y .

Let π denote the projection of X × Y onto X. Then, by Proposition 2.1, eπ(Ui) = dim(Y ).Hence, Ui = Vi × Y for a subvariety Vi ⊂ X for each i. By the projection formula, [Z] =∑ai[Vi]. Since Z is extremal in X, the classes of Z and Vi are proportional. Hence, their

pullbacks to X ×Y are proportional under rational equivalence, and also proportional undernumeral equivalence by our assumption. We thus conclude that [Ui] is proportional to [Z×Y ]for every i and Z × Y is extremal in X × Y . �

Another useful criterion is the following. Let Ak(X) (resp. Ak(X)) denote the Chowgroup of rationally equivalent cycle classes of dimension k (resp. codimension k) in X withR-coefficients. If two cycles are rationally equivalent, they are also numerically equivalent,


hence we have a map Ak(X) → Nk(X). For a morphism Y → X, there is a pushforwardmap Ak(Y )→ Ak(X).

Proposition 2.5. Let γ : Y → X be a morphism between two projective varieties. Assumethat Ak(Y ) → Nk(Y ) is an isomorphism and that the composite γ∗ : Ak(Y ) → Ak(X) →Nk(X) is injective. Moreover, assume that f : X →W is a morphism to a projective varietyW whose exceptional locus is contained in γ(Y ). If a k-dimensional subvariety Z ⊂ Y is anextremal cycle in Effk(Y ) and if ef (γ(Z)) > 0, then γ(Z) is also extremal in Effk(X).

Proof. Suppose that [γ(Z)] = a1[Z1] + · · · + ar[Zr] ∈ Nk(X) for subvarieties Zi ⊂ X withai > 0 for all i. Since ef (γ(Z)) > 0, by Proposition 2.1, Zi drops dimension under f .Hence, Zi is contained in γ(Y ) for all i. Let Z ′i ⊂ Y such that γ∗[Z

′i] = [Zi]. Then γ∗([Z] −

a1[Z ′1]− · · · − ar[Z ′r]) = 0. Since γ∗ is injective, it follows that [Z] = a1[Z ′1] + · · ·+ ar[Z′r] in

Ak(Y ) ∼= Nk(Y ). Since Z is extremal in Effk(Y ), we conclude that [Z ′i] is proportional to [Z]in Nk(Y ). Therefore, [Zi] = γ∗[Z

′i] is proportional to γ∗[Z] in Nk(X) as well. �

We will apply Proposition 2.5 to the case when X is the moduli space of curves and Y isan irreducible boundary divisor. Occasionally, we will be able to prove that effective cyclesexpressing a cycle class [Z] are contained in a union of divisors. We will then use the followingtechnical result to deduce the extremality of [Z].

Proposition 2.6. Let Y =⋃ni=0Di be a union of irreducible divisors Di ⊂ X such that

Di ∩ Dj consists of mi,j irreducible codimension two subvarieties Di,j,k for 0 ≤ i < j ≤ nand k = 1, . . . ,mi,j. Assume for all i, j, k that Di,j,k is extremal in Di and that a linearcombination

n∑j=0j 6=i

mi,j∑k=1

ai,j,k[Di,j,k]

is effective in Di if and only if ai,j,k ≥ 0. Further assume that A1(Di) → N1(Di) is anisomorphism and A1(Y ) → A2(X) → N2(X) are injective. Let Z ⊂ D0 be an effectivedivisor. Finally, assume that for every effective expression

[Z] =∑i

ai[Zi] ∈ Eff2(X),

where ai > 0 and Zi ⊂ X is a codimension two subvariety, Zi is contained in Y for all i. IfZ is extremal in Eff1(D0), then Z is also extremal in Eff2(X).

Proof. Suppose that

[Z] =∑i

ai[Zi] ∈ Eff2(X)(1)

for ai > 0 and Zi ⊂ X irreducible codimension two subvariety. We want to show that [Zi] isproportional to [Z] in N2(X). By assumption, Zi is contained in Y for all i. Reexpress thesummation in (1) as

S0 + · · ·+ Sn, where Sj =∑

Zi⊂Dj ,

Zi 6⊂Dk for k<j

ai[Zi].


By assumption, the second map inn⊕i=1

N1(Di)→ N1(Y )→ N2(X)

is injective. The kernel of the first map is generated by elements of type

(0, · · · , [Di,j,k], · · · ,−[Di,j,k], · · · , 0)

for 0 ≤ i < j ≤ n and 1 ≤ k ≤ mi,j , where the nonzero entries occur in the ith and jthplaces. Lifting (1) to the direct sum, there exist ai,j,k ∈ R such that

[Z] = S0 +n∑j=1

m0,j∑k=1

a0,j,k[D0,j,k] ∈ N1(D0)(2)

and

0 = Si +

i−1∑j=0

mj,i∑k=1

(−aj,i,k)[Dj,i,k] +

n∑j=i+1

mi,j∑k=1

ai,j,k[Di,j,k] ∈ N1(Di)(3)

for 1 ≤ i ≤ n.

In the expressions, each ai,j,k appears twice with opposite signs. Since Si is effective, ourassumption that the coefficients of Di,j,k in an effective sum have to be positive along with(3) implies that aj,i,k ≥ 0 for 0 ≤ j < i ≤ n and ai,j,k ≤ 0 for 1 ≤ i < j ≤ n. Therefore, weconclude that ai,j,k = 0 for 1 ≤ i < j ≤ n and all k. Now (3) reduces to

Si =

m0,j∑k=1

a0,j,k[D0,i,k] ∈ N1(Di)(4)

for 1 ≤ i ≤ n. First, assume that [Z] is not proportional to any [D0,j,k] in N1(D0). Byassumption that Z is extremal in D0 and a0,i,k ≥ 0 for i > 0, (2) and (4) imply thata0,j,k = 0 for all j, k, Si = 0 and the classes [Zi] in S0 are proportional to [Z]. Theclasses [D0,j,k] are independent in D0, otherwise it would contradict the assumption that∑n

j=1

∑m0,j

k=1 a0,j,k[D0,j,k] is effective in D0 if and only if a0,j,k ≥ 0. Therefore, we conclude

that the composite N1(D0)→ N1(Y )→ N2(X) is injective, and hence Z is also extremal inEff2(X). If [Z] is proportional to some [D0,j,k] in N1(D0), then by (2) and (4), a0,j′,k′ = 0for all (j′, k′) 6= (j, k) and classes in the summation Si are proportional to [Z] in N1(Di) forall i ≥ 0. We still conclude that Z is extremal in Eff2(X). �

Remark 2.7. Under the assumptions of Proposition 2.2, one can further show that [Zi] isextremal in the pseudoeffective cone Effk(X). The argument follows from very recent progressin the study of kernels of numerical pushforwards, see [FL2, Section 7]. We briefly explainthe idea. For a morphism f : X → Y of projective varieties, recall that the index ef (Z)associated to a k-dimensional subvariety Z ⊂ X measures the dimension decrease of Z underf . Equivalently, let H be an ample divisor class on Y . Then ef (Z) = 1 + cf (Z), where

cf (Z) is the largest integer c ≤ k such that [Z] · f∗Hk−c = 0. In [FL2], cf (Z) is called thecontractibility index of Z, and can be defined in the same way for any pseudoeffective classin Effk(X).

Returning to Proposition 2.2, there is no subvariety W of X such that dimW > k andef (W ) ≥ k −m. Otherwise there would exist infinitely many k-dimensional subvarieties Zisatisfying ef (Zi) ≥ k−m, contradicting the assumptions. Therefore by [FL2, Theorem 7.18],


for any nonzero pseudoeffective class α ∈ Effd(X) such that cf (α) ≥ k−m−1, we have d ≤ k.

Now suppose that [Zi] =∑

j ajαj such that aj > 0, αj ∈ Effk(X) and not proportional to

[Zi]. It is easy to see that cf (αj) ≥ cf (Zi) = ef (Zi)−1 = k−m−1. By [FL2, Theorem 7.18]again, αj is a nonnegative linear combination of [Z1], . . . , [Zn], hence [Zi] is a nonnegativelinear combination of the other [Zl]’s, contradicting the assumption of linear independenceof their classes. So we have shown that [Zi] is extremal in Effk(X).

In general, without the presence of a contraction morphism (e.g. Propositions 2.5 and 2.6),we do not know how to extend the extremality of an effective higher codimension cycle tothe pseudoeffective cone. This is related to the following subtle and technical question. Letf : X → Y be a morphism and α ∈ Effk(X) such that f∗α = 0. Is α in the closure ofthe cone generated by k-dimensional subvarieties that are contracted by f? The homologicalversion of the question was first raised by [DJV], and was answered affirmatively for curvesand divisors. The numerical analogue of the question was studied extensively in [FL2], and anumber of new cases were established. We refer to these papers for further details. Anotherrelated question is the following. Does there exist a projective variety X and a k-dimensionalsubvariety Z such that [Z] is extremal in Effk(X) but fails to be extremal in Effk(X)? Wedo not know any examples.

3. Preliminaries on moduli spaces of curves

LetMg,n be the moduli space of stable genus g curves with n ordered marked points. The

boundary ∆ ofMg,n consists of irreducible boundary divisors ∆0 and ∆i;S for 0 ≤ i ≤ [g/2],S ⊂ {1, . . . , n} such that |S| ≥ 2 if i = 0. A general point of ∆0 parameterizes an irreduciblenodal curve of geometric genus g − 1. A general point of ∆i;S parameterizes a genus i curvecontaining the marked points labeled by S, attached at one point to a genus g − i curvecontaining the marked points labeled by the complement Sc. We use λ to denote the firstChern class of the Hodge bundle. Let ψi be the first Chern class of the cotangent line bundleassociated to the ith marked point and let ψ =

∑ni=1 ψi.

We may also consider curves with unordered marked points. Let Mg,n =Mg,n/Sn be themoduli space of stable genus g curves with n unordered marked points. The boundary of

Mg,n consists of irreducible boundary divisors ∆0 and ∆i;k for 0 ≤ i ≤ [g/2] and 0 ≤ k ≤ nsuch that k ≥ 2 if i = 0.

Moduli spaces of curves of lower genera can be glued together to form the boundary of

Mg,n. Set ∆0 =Mg−1,n+2/S2, where S2 interchanges the last two marked points. Identify-ing the last two marked points to form a node induces a gluing morphism

α0 : ∆0 → ∆0.

For 0 < i < g/2 or i = g/2 if g is even and n > 0, denote by ∆i;S =Mi,|S|+1×Mg−i,n−|S|+1.Identifying the last marked points in the two factors induces

αi;S : ∆i;S → ∆i;S .

For g even, i = g/2 and n = 0, set ∆g/2 = (Mg/2,1 ×Mg/2,1)/S2 and

αg/2 : ∆g/2 → ∆g/2

is induced by identifying the two marked points to form a node. Restricted to the complementof the locus of curves with more than one node, the gluing morphisms are isomorphisms. Later


on when studying codimension two boundary strata of Mg,n, this will help us identify themwith boundary divisors on moduli spaces of curves of lower genera.

In order to study extremal higher codimension cycles on Mg,n and Mg,n, we need to usethe fact that the codimension one boundary strata are extremal. This is well-known (see e.g.[R1, 1.4]), and we explain it briefly as follows.

Proposition 3.1. Every irreducible boundary divisor is extremal on Mg,n and Mg,n.

Proof. In general, one can exhibit a moving curve in a boundary divisor that has negativeintersection with the divisor, which implies that it is extremal by [CC2, Lemma 4.1]. For ∆0,fix a curve C of genus g − 1 with n + 1 distinct fixed points p1, . . . , pn+1. One obtains thedesired moving curve by gluing a varying point on C to pn+1. For ∆i;S , fix a curve C of genusi with distinct marked points labeled by S and a curve C ′ of genus g− i with distinct markedpoints labeled by Sc. One obtains the desired moving curve by gluing a fixed point on C ′

distinct from the previously chosen points to a varying point on C. The only exceptionalcase is ∆0 in g = 1. Its class equals 12λ, which is semi-ample. However, λ induces a fibrationover M1,1, hence it spans an extremal ray in both the nef cone and the effective cone. �

We also need to use several other compactifications ofMg,n. Let τ :Mg → Asatg denote the

extended Torelli map from Mg to the Satake compactification of the moduli space Ag of g-dimensional principally polarized abelian varieties (see e.g. [BHPV, III. 16]). It maps a stablecurve to the product of the Jacobians of the irreducible components of its normalization. Theexceptional locus of τ is the total boundary ∆.

Let ps :Mg,n →Mpsg,n be the first divisorial contraction of the log minimal model program

forMg,n ([HH, AFSV]), whereMpsg,n is the moduli space of genus g pseudostable curves with

n ordered marked points. It contracts ∆1;∅ only, replacing an unmarked elliptic tail by acusp.

Let A = {a1, . . . , an} be a collection of n rational numbers such that 0 < ai ≤ 1 for all iand 2g − 2 +

∑ni=1 ai > 0. The moduli space of weighted stable curves Mg,A parameterizes

the data (C, p1, . . . , pn,A) such that

• C is a connected, reduced, at-worst-nodal arithmetic genus g curve.• p1, . . . , pn are smooth points of C assigned the weights a1, . . . , an, respectively, where

the total weight of any points that coincide is at most one.• for every irreducible component X of C, the divisor class KC +

∑ni=1 aipi is ample

restricted to X, i.e. numerically

2gX − 2 + #(X ∩ C\X) +∑pi∈X

ai > 0,

where gX is the arithmetic genus of X.

If A = {1, . . . , 1}, then Mg,A = Mg,n. Hassett constructs a morphism fA : Mg,n → Mg,Athat modifies the locus of curves violating the stability inequality in the above [Hass]. Note

that if gX ≥ 1 or #(X∩C\X) ≥ 2 for C parameterized byMg,n, the above inequality alwaysholds for any permissible A. Hence, the exceptional locus of fA consists of curves that havea rational tail with at least three marked points of total weight at most one.


4. The effective cones of M3 and M3,1

We want to study the effective cone of higher codimension cycles onMg,n. First, consider

the case of codimension two and n = 0. Since dimM2 = 3, a cycle of codimension two onM2 is a curve, and the cone of curves ofM2 is known to be spanned by the one-dimensionaltopological strata (F -curves). In this section, we introduce our methods by studying the firstinteresting case g = 3 in detail. In later sections, we will generalize some of the results tog ≥ 4 or n > 0.

The Chow ring of M3 was computed in [Fab1]. For ease of reference, we preserve Faber’snotation introduced in [Fab1, p. 340–343]:

• Let ∆00 ⊂M3 be the closure of the locus parameterizing irreducible curves with twonodes.• Let ∆01a ⊂ M3 be the closure of the locus parameterizing a rational nodal curve

attached to a genus two curve at one point.• Let ∆01b ⊂ M3 be the closure of the locus parameterizing an elliptic nodal curve

attached to an elliptic curve at one point.• Let ∆11 ⊂ M3 be the closure of the locus parameterizing a chain of three elliptic

curves.• Let Ξ0 ⊂ M3 be the closure of the locus parameterizing irreducible nodal curves

in which the normalization of the node consists of two conjugate points under thehyperelliptic involution.• Let Ξ1 ⊂M3 be the closure of the locus parameterizing two elliptic curves attached

at two points.• Let H1 ⊂M3 be the closure of the locus of curves consisting of an elliptic tail attached

to a genus two curve at a Weierstrass point.

The codimension two boundary strata of M3 consist of ∆00, ∆01a, ∆01b, ∆11 and Ξ1. Wedenote the cycle class of a locus by the corresponding small letter, such as δ00 for the classof ∆00. By [Fab1], the Chow group A2(M3) is isomorphic to N2(M3), with a basis given byδ00, δ01a, δ01b, δ11, ξ0, ξ1 and h1 over R.

The interior of ∆0 parameterizing irreducible curves with exactly one node is given by

Int ∆0 = ∆0 −∆00 −∆01a −∆01b − Ξ1.

Faber shows that A1(Int ∆0) is generated by ξ0 ([Fab1, Lemma 1.12]). In particular, A1(∆0)is generated by δ00, δ01a, δ01b, ξ0 and ξ1, and A1(∆0)→ N2(M3) is injective.

The interior of ∆1 parameterizing the union of a smooth genus one curve and a smoothgenus two curve attached at one point, is given by

Int ∆1 = ∆1 −∆01a −∆01b −∆11.

Faber shows that A1(Int ∆1) is generated by h1 ([Fab1, Lemma 1.11]). In particular, A1(∆1)is generated by δ01a, δ01b, δ11 and h1, and A1(∆1)→ N2(M3) is injective.

Before studying effective cycles of codimension two in M3, we need to study effectivedivisors in ∆0 and ∆1. Recall the gluing maps

α0 : ∆0 = M2,2 → ∆0,

α1 : ∆1 =M1,1 ×M2,1 → ∆1.


By the divisor theory ofMg,n and Mg,n, we know that A1(∆0) = N1(∆0) is generated by the

classes of inverse images of ∆00, ∆01a, ∆01b, Ξ0 and Ξ1 under α0. Similarly, A1(∆1) = N1(∆1)is generated by the classes of inverse images of ∆01a, ∆01b, ∆11 andH1 under α1. In particular,

(ι ◦ αi)∗ : A1(∆i)→ A2(M3) is injective for i = 0, 1, where ι : ∆ ↪→M3 is the inclusion. Forease of notation, we denote the inverse image of a cycle class under αi by the same symbol.

Lemma 4.1. The classes δ01a, δ01b, δ11 and h1 are extremal in Eff1(∆1).

Proof. The effective cone of divisors on M2,1 is computed in [R1, Section 3.3]. Note that

δ01a corresponds to the fiber class of ∆1 over M1,1, while δ01b, δ11 and h1 are the pullbacks

of the extremal divisor classes δ0, δ1 and w from M2,1, respectively, where w is the divisor

class of the locus of curves with a marked Weierstrass point in M2,1. By Corollary 2.4, δ01a,δ01b, δ11 and h1 are extremal. �

Let BN12 ⊂ M2,2 be the closure of the locus of curves such that the two unordered marked

points are conjugate under the hyperelliptic involution. It has divisor class

[BN12 ] = −1

2λ+

1

2ψ − 3

2δ0;2 − δ1;0,

see e.g. [Lo].

Lemma 4.2. The classes δ0, δ0;2, δ1;0, δ1;1 and [BN12 ] are extremal in Eff1(∆0). Consequently

δ00, δ01a, δ01b, ξ1 and ξ0 are extremal as classes in Eff1(∆0).

Proof. By Proposition 3.1, the boundary divisors are known to be extremal. To obtain amoving curve B in BN1

2 that has negative intersection with it, fix a general genus two curveC and vary a pair of conjugate points (p1, p2) in C. Since [B] ·λ = 0, [B] ·ψ = 16, [B] ·δ0;2 = 6and [B] ·δ1;0 = 0, we have [B] · [BN1

2 ] < 0. Hence, BN12 is extremal. Note that the classes δ00,

δ01a, δ01b, ξ1 and ξ0 in ∆0 correspond to δ0, δ0;2, δ1;0, δ1;1 and [BN12 ] in M2,2, respectively,

thus proving the claim. �

Theorem 4.3. The classes δ00, δ01a, δ01b, ξ1, ξ0, δ11 and h1 are extremal in Eff2(M3).

Proof. Recall the Torelli map τ :Mg → Asatg and the first divisorial contraction ps :Mg →

Mpsg discussed in Section 3. For g = 3, note that ∆01b, ∆11 and H1 are contained in ∆1,

and eps(∆01b), eps(∆11), eps(H1) > 0. Moreover, by Lemma 4.1 their classes are extremal in

Eff1(M1,1 ×M2,1). Proposition 2.5 implies that they are extremal in Eff2(M3).

The strata ∆00 and Ξ1 are contained in ∆0 and have index eτ ≥ 2. If Z is a subvarietyof codimension two in M3 with eτ (Z) ≥ 2, then Z is either contained in ∆0 or in ∆1. Sinceeτ (∆1) = 1, if Z is contained in ∆1 but not in ∆0, then Z has to be ∆11. By Lemma 4.2,

δ00 and ξ1 are extremal in Eff1(M2,2). Moreover, δ00, ξ1 and δ11 are linearly independent in

N2(M3). We thus conclude that δ00 and ξ1 are extremal in Eff2(M3) by Proposition 2.5.

The remaining cases are ∆01a and Ξ0. A rational nodal tail does not have moduli. Con-sequently, eτ (∆01a) = 1 and eps(∆01a) = 0. Similarly we have eτ (Ξ0) = 1 and eps(Ξ0) = 0.The dimension drops are too small to directly apply Proposition 2.2. Nevertheless, theireτ indices are positive, hence we may apply Proposition 2.6 to Y = ∆0 ∪ ∆1. Recall that

N1(∆i) ∼= A1(∆i) → N2(M3) is injective for i = 0, 1. Moreover, ∆0 ∩ ∆1 = ∆01a ∪ ∆01b

and both components are extremal in ∆0 and in ∆1 by Lemmas 4.1 and 4.2. Finally, ξ0


is extremal in ∆0 by Lemma 4.2. By Proposition 2.6, we conclude that ∆01a and Ξ0 areextremal in Eff2(M3). �

Remark 4.4. It would be interesting to find an extremal cycle of codimension two that isnot contained in the boundary ofM3. Some natural geometric non-boundary cycles fail to beextremal. For instance, the closure B3 of the locus of bielliptic curves inM3 is not extremal.The class of B3 was calculated in terms of another basis of A2(M3) ([FP]):

[B3] =2673

2λ2 − 267λδ0 − 651λδ1 +

27

2δ2

0 + 69δ0δ1 +177

2δ2

1 −9

2κ2.

By [Fab1, Theorem 2.10], we can rewrite it as follows:

[B3] =1

2δ00 +

25

8δ01a +

19

4δ01b + 18δ11 +

15

8ξ0 + 15ξ1 + 10h1.

Therefore, B3 is not extremal in Eff2(M3).

However, if we consider M3,1 instead, we are able to find an extremal non-boundarycodimension two cycle. First, let us introduce some notation:

• Let H ⊂M3 be the closure of the locus of hyperelliptic curves.• Let HP ⊂ M3,1 be the closure of the locus of hyperelliptic curves with a marked

point.• Let HW ⊂ M3,1 be the closure of the locus of hyperelliptic curves with a marked

Weierstrass point.

Clearly HW is a subvariety of codimension two in M3,1, which is not contained in theboundary. Moreover, HW ⊂ HP = π−1(H) and π : HW → H is generically finite of degreeeight, where π :M3,1 →M3 is the morphism forgetting the marked point.

We will show that HW is an extremal cycle in M3,1. In order to prove this, we need tounderstand the divisor theory of HP as well as the boundary components ∆1;{1} and ∆1;∅ of

M3,1. We further introduce the following cycles:

• Let HP0 ⊂ HP be the closure of the locus of irreducible nodal hyperelliptic curveswith a marked point.• Let HP1 = HP ∩∆1;{1} be the closure of the locus of curves consisting of a marked

genus one component attached to a genus two component at a Weierstrass point.• Let HP2 = HP ∩∆1;∅ be the closure of the locus of curves consisting of a genus one

component attached at a Weierstrass point to a marked genus two component.• Let Θ ⊂ HP be the closure of the locus of two genus one curves attached at two

points, one of the curves marked.• Let D01a ⊂ ∆1;{1} be the closure of the locus of a marked genus one curve attached

to an irreducible nodal curve of geometric genus one.• Let D02b ⊂ ∆1;{1} be the closure of the locus of a marked rational nodal curve attached

to a genus two curve.• Let D12 ⊂ ∆1;{1}∩∆1;∅ be the closure of the locus of a chain of three curves of genera

2, 0 and 1, respectively, such that the rational component contains the marked point.• Let D11a ⊂ ∆1;{1} ∩ ∆1;∅ be the closure of the locus of a chain of three genus one

curves such that one of the two tails contains the marked point.• Let W1 ⊂ ∆1;∅ be the closure of the locus of a genus one curve attached to a marked

genus two curve such that the marked point is a Weierstrass point.


• Let D02a ⊂ ∆1;∅ be the closure of the locus of a rational nodal curve attached to amarked genus two curve.• Let D01b ⊂ ∆1;∅ be the closure of the locus of a genus one curve attached to a marked

irreducible nodal curve of geometric genus one.• Let D11b ⊂ ∆1;∅ be the closure of the locus of a chain of three genus one curves such

that the middle component contains the marked point.

Recall the gluing morphisms

α1;{1} : ∆1;{1} =M1,2 ×M2,1 → ∆1;{1},

α1;∅ : ∆1;∅ =M1,1 ×M2,2 → ∆1;∅.

For ease of notation, we denote the inverse image of a class under the gluing maps by thesame symbol.

Lemma 4.5. (i) N1(HP ) ∼= A1(HP ) is generated by hw, hp0, hp1, hp2 and θ.(ii) The divisor classes hw, hp0, hp1, hp2 and θ are extremal in HP . Moreover, if a

linear combination a1 · hp1 + a2 · hp2 is effective, then a1, a2 ≥ 0.

(iii) N1(∆1;{1}) ∼= A1(∆1;{1}) is generated by hp1, d01a, d02b, d11a and d12. Moreover, if alinear combination b1 · hp1 + b2 · d11a + b3 · d12 is effective, then bi ≥ 0 for all i.

(iv) N1(∆1;∅) ∼= A1(∆1;∅) is generated by hp2, w1, d01b, d02a, d11a, d11b and d12. Moreover,if a linear combination c1 · hp2 + c2 · d11a + c3 · d12 is effective, then ci ≥ 0 for all i.

(v) The kernel of N1(HP )⊕N1(∆1;{1})⊕N1(∆1;∅)→ N2(M3,1) is generated by

(hp1,−hp1, 0), (hp2, 0,−hp2), (0, d11a,−d11a) and (0, d12,−d12).

Proof. The first two claims essentially follow from [R2, 9.2, 10.1]. A curve parameterized inHP is a genus three hyperelliptic curve C with a marked point p. Hence, it can be identifiedwith an admissible double cover ([HMo, 3.G]) of a stable rational curve R branched at eightunordered points q1, . . . , q8 with a distinguished marked point q as the image of p. Markingthe conjugate point p′ of p gives (C, p′), which is isomorphic to (C, p) under the hyperellipticinvolution. Therefore, the data (R, q1, . . . , q8, q) uniquely determines (C, p). The curve (C, p)is not stable if and only if R has a rational tail marked by exactly two of the unordered pointsor by an unordered point and q or has a rational bridge marked only by q. In the first case,the double cover has an unmarked rational bridge. In the second case, the double cover hasa rational tail marked by q. In the last case, the double cover has a rational bridge with nomarked points. In all three cases stabilization contracts the nonstable rational component.Furthermore, when R has a rational tail marked by exactly two unordered points and q, thedouble cover has a rational bridge marked by q. While a rational bridge marked by one pointhas no moduli, a rational tail with three marked points has one-dimensional moduli.

Following the notation in [R2], let X9,1 = M0,9/S8 be the moduli space of stable genuszero curves with nine marked points q1, . . . , q8, q such that q1, . . . , q8 are unordered but q isdistinguished. Let Bk ⊂ X9,1 be the boundary divisor parameterizing curves with a rationaltail marked by q and k−1 of the qi for 2 ≤ k ≤ 7. There is a morphism X9,1 → HP defined bytaking the admissible double cover and stabilizing. This morphism contracts the boundarydivisor B3 to the locus of hyperelliptic curves with a marked rational bridge. Numericalequivalence, rational equivalence and linear equivalence over R coincide for divisors on modulispaces of stable pointed genus zero curves, and the Picard group is generated by boundarydivisors. Considering the admissible double covers corresponding to the boundary divisorsB2, B4, B5, B6 and B7 of X9,1, we see that their images in HP correspond to HW , HP1, Θ,


HP2 and HP0, respectively, hence (i) follows. The same curves proving the extremality ofthe boundary divisors in this case (see [R2, 10.1]) also prove (ii).

For (iii), hp1 and dij correspond to the pullbacks of generators of N1(M1,2) ∼= A1(M1,2)

and of N1(M2,1) ∼= A1(M2,1), where ∆1;{1} =M1,2 ×M2,1. Moreover, the interior M1,2 ×M2,1 is affine. Hence, these classes generate N1(∆1;{1}) ∼= A1(∆1;{1}). Suppose that b1 ·hp1 +

b2 · d11a + b3d12 is an effective divisor class. Let π1 and π2 be the projections of ∆1;{1} to

M1,2 andM2,1, respectively. Then, we have hp1 = π∗2w, d11a = π∗2δ1;{1} and d12 = π∗1δ0;{1,2},where w is the divisor class of Weierstrass points. Hence, b3 ≥ 0 and b1 · w + b2 · δ1;{1} is

effective in M2,1. Since w and δ1;{1} span a face of Eff1(M2,1) ([R1, Corollary 3.3.2]), weconclude that b1, b2 ≥ 0. A similar argument yields (iv).

For (v), since HP ∩∆1;{1} = HP1, HP ∩∆1;∅ = HP2 and ∆1;{1} ∩∆1;∅ = D12 ∪D11a, itsuffices to prove that hw, hp0, hp1, hp2, θ, w1, d01a, d01b, d02a, d02b, d11a, d11b and d12 areindependent in N2(M3,1). Suppose that they satisfy a relation

a · hw + t · θ + s · w1 +∑i

bi · hpi +∑i,j,k

cijk · dijk = 0 ∈ N2(M3,1)(5)

for a, bi, cijk, s, t ∈ R. The morphism π : M3,1 → M3 contracts these cycles except HW ,W1 and D12, where π(HW ) = H and π(W1) = π(D12) = ∆1. Since π is flat and h, δ1 areindependent in M3, applying π∗ to (5) we conclude that a = 0.

For the remaining cycles, we have π(HP1) = π(HP2) = H1, π(HP0) = Ξ0, π(Θ) = Ξ1,π(D02a) = π(D02b) = ∆01a, π(D01a) = π(D01b) = ∆01b and π(D11a) = π(D11b) = ∆11. Theimages of W1 and D12 are contained in ∆1. By [Fab1], the subspace spanned by ξ0 and ξ1

has zero intersection with A1(∆1) in N2(M3). Intersect (5) with an ample divisor class andapply π∗. We thus conclude that t = b0 = 0.

Relation (5) reduces to

s · w1 + b1 · hp1 + b2 · hp2 +∑i,j,k

cijk · dijk = 0.(6)

Recall the pseudostable map ps :M3,1 →Mps3,1. Applying ps∗, we obtain that

b1 · (ps∗ hp1) + c01a · (ps∗ d01a) + c02a · (ps∗ d02a) + c02b · (ps∗ d02b) = 0(7)

in N2(Mps3,1), since the other summands in (6) are contracted by ps. Consider a family S1

of plane quartics passing through 12 general points. Marking one of the points, S1 gives riseto a general two-dimensional family of plane quartics with a marked base point. Then S1

intersects ps(D02a) at finitely many points parameterizing cuspidal quartics and S1 does notintersect the other cycles in (7), which implies that c02a = 0. Let πps :Mps

3,1 →Mps3 be the

morphism forgetting the marked point and pseudostablizing. Note that eπps(ps(D02b)) = 1

and eπps(ps(HP1)) = eπps(ps(D01a)) = 2. Take an ample divisor class in Mps3,1, intersect (7)

and apply πps ∗. We thus conclude that c02b = 0. Finally, take a pencil of plane cubics,mark a base point, take another base point and use it to attach the pencil to a varyingpoint in a fixed genus two curve. We obtain a two-dimensional family S2 in Mps

3,1 such that[S2] · (ps∗ hp1) = −6 and [S2] · (ps∗ d01a) = 0, which implies that b1 = c01a = 0.

Relation (5) further reduces to

s · w1 + b2 · hp2 + c12 · d12 + c11a · d11a + c01b · d01b + c11b · d11b = 0.(8)


Inside the parameter space P14 of plane quartics, take a general two-dimensional subspaceS3. As S3 is general, it contains finitely many general cuspidal curves. Take a line L passingthrough a Weierstrass point of the normalization of one of the cuspidal quartics C and makeL general other than that. Mark the intersection points of L with curves in S3, make a basechange, perform stable reduction and still denote by S3 the resulting family in M3,1. Notethat S3 does not intersect the summands in (8) except W1, and it intersects W1 along aone-dimensional locus parameterizing the normalization of C with a pencil of elliptic tailsattached at the inverse image of the cusp of C, which implies that [S3] · w1 < 0 since thenormal bundle of elliptic tails in ∆1 has negative degree restricted to this family. We thusconclude that s = 0.

Set s = 0, intersect (8) with the ψ-class of the marked point and apply π∗. Since themarked point lies in the rational bridge for curves in D12, it implies that ψ ·d12 = 0. We thusobtain that

(3b2) · h1 + (c11a + 2c11b) · δ11 + c01b · δ01b = 0

in N2(M3). Since h1, δ11 and δ01b are independent in M3 ([Fab1]), we conclude that b2 =c01b = 0 and c11a + 2c11b = 0.

Now Relation (5) reduces to

c12 · d12 − 2c11b · d11a + c11b · d11b = 0.(9)

Attach two pencils of plane cubics to a smooth genus one curve E at two general points, andmark a third general point in E. We obtain a two-dimensional family S4 in M3,1 such that[S4] · d12 = [S4] · d11a = 0 and [S4] · d11b = 1. Plugging in (9), we obtain that c11b = 0, hencec12 · d12 = 0 and c12 = 0. �

Now we are ready to prove that HW is extremal in M3,1. In fact, all extremal divisors in

HP are extremal as codimension two cycles in M3,1. First, observe that π :M3,1 →M3 is

flat of relative dimension one. As a consequence, if Z ⊂M3,1 is an irreducible subvariety of

codimension two, then π(Z) ⊂M3 has codimension either one or two, i.e. eπ(Z) = 0 or 1.

Theorem 4.6. The cycle classes of HW , HP0, HP1, HP2 and Θ are extremal in Eff2(M3,1).

Proof. We prove it for HW first. Suppose that

hw =

r∑i=1

ai[Yi] +

s∑j=1

bj [Zj ] ∈ N2(M3,1)(10)

where ai, bj > 0 and Yi, Zj ⊂M3,1 are irreducible subvarieties of codimension two such thateπ(Yi) = 0 and eπ(Zj) = 1. Since π∗[Zj ] = 0, we have

8 · h =r∑i=1

ai · π∗[Yi] ∈ N1(M3).

Since H is an extremal and rigid divisor inM3 (see e.g. [R1, 2.2]), it follows that π(Yi) = H.Hence, Yi ⊂ π−1(H) = HP for all i.

Next, we show that Zj is contained in HP ∪ ∆1;{1} ∪ ∆1;∅ for all j. Consider a familyS of plane quartics passing through twelve general points. Marking one of the points, Scan be regarded as a two-dimensional family of plane quartics with a marked base point.By taking the base points general enough, we can assume that S avoids any phenomenonof codimension three or higher. In particular, we may assume that S does not contain any


reducible or nonreduced elements and S has finitely many cuspidal elements such that thebase point is not any of the cusps. Furthermore, we may specify S to contain a specificsmooth or non-hyperelliptic irreducible nodal quartic in Z1 while preserving these properties.Then S induces a surface in M3,1 such that [S] · hw = 0, [S] · [Yi] ≥ 0, [S] · [Zj ] ≥ 0 forj 6= 1 and [S] · [Z1] > 0. Intersecting both sides of (10) with S leads to a contradiction. Wethus conclude that Yi, Zj ⊂ HP ∪ ∆1;{1} ∪ ∆1;∅ for all i, j. By Lemma 4.5, we can apply

Proposition 2.6 to Y = HP ∪∆1;{1} ∪∆1;∅, thus proving that hw is extremal in Eff2(M3,1).

For HP0, HP1, HP2 and Θ, note that their eπ indices are all equal to one and their imagesunder π are contained in H. If any of them satisfies a relation like (10), by Proposition 2.1the Yi terms cannot exist in the effective expression. Then the same argument in the previousparagraph implies that the remaining terms Zj are contained in HP ∪∆1;{1} ∪∆1;∅ for all j.Hence, we can apply Lemma 4.5 and Proposition 2.6 to conclude that they are extremal inEff2(M3,1). �

5. The effective cones of Mg for g ≥ 4

In this section, we study effective cycles of codimension two onMg for g ≥ 4. The method

is similar to the case g = 3. There is a topological stratification of Mg, where the strata areindexed by dual graphs of stable nodal curves. The codimension two boundary strata consistof curves with at least two nodes. We recall the notation introduced in [E] for these strata:

• Let ∆00 be the closure of the locus inMg parameterizing irreducible curves with twonodes.• For 1 ≤ i ≤ j ≤ g − 2 and i + j ≤ g − 1, let ∆ij be the closure of the locus in Mg

parameterizing a chain of three curves of genus i, g − i− j and j, respectively.• For 1 ≤ j ≤ g−1, let ∆0j be the closure of the locus inMg parameterizing a union of

a genus j curve and an irreducible nodal curve of geometric genus g− 1− j, attachedat one point.• For 1 ≤ i ≤ [(g − 1)/2], let Θi be the closure of the locus in Mg parameterizing a

union of a curve of genus i and a curve of genus g − i− 1, attached at two points.

In order to study higher codimension cycles on Mg, we need to understand the divisor

theory of the boundary components. Recall the gluing morphisms αi : ∆i → ∆i for 0 ≤i ≤ [g/2]. By [Fab3, p. 69] and [E, Section 4], A1(∆) → N2(Mg) is injective. Moreover,

(ι ◦ αi)∗ : N1(∆i) ∼= A1(∆i) → N2(Mg) is injective, where ι : ∆ ↪→Mg is the inclusion. As

before, we denote the class of a locus in ∆i and in ∆i by the same symbol. Whenever we useg/2 as an index, the corresponding term exists if and only if g is even.

Lemma 5.1. (i) The classes δ11 and δ1g−2 are extremal in Eff1(∆1).

(ii) The class δ0g−1 is extremal in Eff1(∆0).

Proof. Note that ∆1 = M1,1 × Mg−1,1. The classes δ11 and δ1g−2 are the pullbacks of

boundary classes δ1; and δg−2; fromMg−1,1, respectively. Now (i) follows from Corollary 2.4and Proposition 3.1.

For (ii), δ0g−1 corresponds to the boundary class δ0;2 in ∆0 = Mg−1,2. Hence, the claimfollows from Proposition 3.1. �

Lemma 5.2. (i) The linear combination a0δ0 +∑g−1

i=1 aiδi,1 is effective on Mg,1 if andonly if all the coefficients are nonnegative.


(ii) For 0 < i < g/2, the linear combination

i−1∑k=0

(ckiδki + dkg−iδkg−i) +

[g/2]∑k=i+1

(cikδik + dig−kδig−k)

is effective on ∆i if and only if all the coefficients are nonnegative.(iii) For g even and i = g/2, the linear combination

g/2−1∑k=0

bkg/2δkg/2

is effective on ∆g/2 if and only if all the coefficients are nonnegative.

Proof. For (i), take a nodal curve of geometric genus g − 1 and vary a marked point on it.We obtain a curve C0 moving in ∆0 such that [C0] · δ0 < 0 and [C0] · δi,1 = 0 for all i. For1 ≤ i ≤ g − 2, sliding a genus i curve with a marked point along a genus g − i curve, weobtain a curve Ci moving in ∆i,1 such that [Ci] · δi,1 < 0, [Ci] · δ0 = 0 and [Ci] · δj,1 = 0 forj 6= i. Finally, attach a genus g − 1 curve with a marked point to a genus one curve at ageneral point and vary the marked point in the component of genus g−1. We obtain a curveCg−1 moving in ∆g−1,1 such that [Cg−1] · δ0 = 0, [Cg−1] · δ1,1 = 1, [Cg−1] · δg−1,1 = 4− 2g < 0and [Cg−1] · δi,1 = 0 for 2 ≤ i ≤ g − 2.

Suppose that D is an effective divisor on Mg,1 with class [D] = a0δ0 +∑g−1

i=1 aiδi,1. Ifthe support of D is contained in the union of ∆0 and the ∆i,1, we are done, because theboundary divisor classes are linearly independent. Suppose that D does not contain anyboundary components in its support. Since the curves constructed are moving in the respec-tive boundary components, we have [D] · [Ci] ≥ 0 for 0 ≤ i ≤ g − 1. We thus conclude thata0 ≤ 0, ai ≤ 0 for 1 ≤ i ≤ g − 2 and (4 − 2g)ag−1 + a1 ≥ 0, hence ag−1 ≤ 0. It follows that

[D] + (−a0)δ0 +∑g−1

i=1 (−ai)δi,1 = 0, leading to a contradiction, because the class of a positivesum of effective divisors cannot be zero. Indeed we have proved that an effective divisor withclass a0δ0 +

∑g−1i=1 aiδi,1 has to be supported in the union of ∆0 and the ∆i,1.

For (ii) and (iii), the boundary classes in the linear combination are the pullbacks of theboundary classes from Mi,1 and Mg−i,1, respectively, hence the claims follow from (i). �

Theorem 5.3. For g ≥ 2, every codimension two boundary stratum of Mg is extremal in

Eff2(Mg).

Proof. When g = 2, the two F -curves ∆00 and ∆01 are dual to the two nef divisors λ and12λ− δ, respectively, and form the extremal rays of the Mori cone of curves. The case g = 3is covered by Theorem 4.3. Hence, we may assume that g ≥ 4.

Recall that the extended Torelli map τ :Mg → Asatg sends a stable curve to the product

of the Jacobians of the irreducible components of its normalization. Suppose that Z is acodimension two boundary stratum of Mg such that a general curve C parameterized by Zdoes not have an irreducible component of geometric genus less than two. This assumptionensures that selecting two points (as the inverse image of a node) in the normalization of Chas two-dimensional moduli. We thus conclude that

eτ (Z) = dimZ − dim τ(Z) = 2 + 2 = 4,


because τ(C1) = τ(C2) if C1 and C2 have the same normalization. If a general curve Cparameterized by Z contains exactly one irreducible component E of geometric genus oneand no other components of geometric genus less than two, then eτ (Z) = 3 since choosing npoints in the normalization of E has (n− 1)-dimensional moduli.

Conversely, if Z ′ ⊂ Mg is a subvariety of codimension two such that eτ (Z ′) ≥ 3, thenevery curve parameterized by Z ′ has at least two nodes. Therefore, Z ′ has to be one of thecodimension two boundary strata. Since the classes of the codimension two boundary strataofMg are independent ([E, Theorem 4.1]), we conclude that [Z] is extremal in Eff2(Mg) byProposition 2.2.

The remaining boundary strata are: ∆11 whose general point parameterizes a chain ofthree curves of genera 1, g − 2 and 1; ∆1g−2 whose general point parameterizes a chain ofthree curves of genera 1, 1 and g − 2; ∆0g−1 whose general point parameterizes a curve witha rational nodal tail. They are all contained in the boundary divisor ∆1.

Recall that ps : Mg →Mpsg contracts the boundary divisor ∆1 by replacing elliptic tails

by cusps. A subvariety of Mg contracted by ps has to be contained in ∆1. Note that

eps(∆11), eps(∆1g−2) > 0, and by Lemma 5.1 (i), δ11 and δ1g−2 are extremal in Eff1(∆1).

Moreover, N1(∆1) ∼= A1(∆1) → N2(Mg) is injective. Therefore, we conclude that δ11 and

δ1g−2 are extremal in Eff2(Mg) by Proposition 2.5.

Finally, for ∆0g−1, the argument is similar to that of ∆01a in the proof of Theorem 4.3.Since eτ (∆0g−1) > 0, we can apply Proposition 2.6 to the setting Y = ∆0 ∪ · · · ∪ ∆[g/2]

and D0 = ∆0. Note that ∆i ∩ ∆j = ∆ij ∪ ∆ig−j and ∆i ∩ ∆g/2 = ∆ig/2 for 0 ≤ i < g/2.Lemmas 5.1 (ii) and 5.2 ensure that Proposition 2.6 applies in this case, thus proving that∆0g−1 is extremal in Eff2(Mg). �

Remark 5.4. If Z is a codimension two subvariety of Mg such that eτ (Z) ≥ 3, the aboveproof shows that Z has to be a codimension two boundary stratum. As a consequenceof Remark 2.7, any codimension two boundary stratum with eτ ≥ 3 is extremal in the

pseudoeffective cone Eff2(Mg).

The techniques of Theorem 5.3 allow us to show that certain boundary strata of arbitrarilyhigh codimension are extremal. In the next theorem, we will give the simplest examples.Given a stable dual graph Γ of arithmetic genus g, let ∆Γ denote the closure of the stratumindexed by Γ in the topological stratification of Mg. Denote the class of ∆Γ by δΓ. Thecodimension of ∆Γ is the number of edges in Γ.

Let κ(g, r) denote the set of stable dual graphs of arithmetic genus g with r edges suchthat the geometric genera of the curves associated to each node is at least two. Let κ′(g, r)be the set of stable dual graphs of arithmetic genus g with r edges such that the geometricgenera of the curves associated to all the nodes but one is at least two and the remaininggenus is at least one. Clearly κ(g, r) ⊂ κ′(g, r). With this notation, we have the followingapplication of Propositions 2.1 and 2.2.

Lemma 5.5. Let Γ ∈ κ(g, r) (resp. Γ ∈ κ′(g, r)). If the class δΓ is not in the span of theclasses δΞ for Γ 6= Ξ ∈ κ(g, r) (resp. Γ 6= Ξ ∈ κ′(g, r)), then δΓ is an extremal cycle ofcodimension r in Mg.

Proof. If Γ ∈ κ(g, r), then eτ (∆Γ) = 2r. If Γ ∈ κ′(g, r) − κ(g, r), then eτ (∆Γ) = 2r − 1.Conversely, if Z ⊂Mg is an irreducible variety of codimension r such that eτ (Z) ≥ 2r, thenZ = ∆Γ for some Γ ∈ κ(g, r). Similarly, if eτ (Z) ≥ 2r−1, then Z = ∆Γ for some Γ ∈ κ′(g, r).


By Proposition 2.1, any effective linear combination expressing δΓ for Γ ∈ κ(g, r) (resp.κ′(g, r)) must be of the form

∑aΞδΞ with Ξ ∈ κ(g, r) (resp. κ′(g, r)). Since δΓ is not in the

span of δΞ for Γ 6= Ξ ∈ κ(g, r) (resp. κ′(g, r)), all the coefficients except for aΓ have to bezero. Therefore, δΓ is an extremal cycle of codimension r. �

In view of Lemma 5.5, it is interesting to determine when the classes δΓ are indepen-dent. Using test families, we give examples of independent classes, which by Lemma 5.5 areextremal.

Let Tr(g) ⊂ κ(g, r) be the set of dual graphs whose underlying abstract graph is a tree withr leaves. If Γ ∈ Tr(g), then the general point of ∆Γ parameterizes curves with r componentsC1, . . . , Cr each attached at one point to distinct points on a curve C0. In addition, each ofthe curves Ci for 0 ≤ i ≤ r have genus at least 2. Let T ′r(g) ⊂ κ′(g, r) be the set of dualgraphs whose underlying abstract graph is a tree with r leaves and the central componentC0 has genus one. Let T ′′r (g) ⊂ κ′(g, r) be the set of dual graphs whose underlying abstractgraph is a tree with r leaves, one of the components Ci, i 6= 0, has genus one and the genusg0 of the central component satisfies 4g0 + 9 ≥ r.

Let Lr−1,g0(g) ⊂ κ(g, r) denote the set of dual graphs whose underlying abstract graphconsists of a vertex v0 with a self-loop and r − 1 leaves, where v0 is assigned a curve ofgeometric genus g0 satisfying 4g0 + 9 ≥ r. If Γ ∈ Lr−1,g0(g), then the general point of ∆Γ

parameterizes curves with r − 1 components C1, . . . , Cr−1 attached at one point to distinctsmooth points on a one-nodal central curve C0 (corresponding to v0) of geometric genus g0.

Theorem 5.6. If Γ is a dual graph in Tr(g)∪T ′r(g)∪T ′′r (g)∪Lr−1,g0(g), then δΓ is independent

from all the classes δΞ for Γ 6= Ξ ∈ κ′(g, r) and δΓ is an extremal codimension r class in Mg.

Proof. Suppose there exists a linear relation

(11)∑

Ξ∈κ′(g,r)

cΞδΞ = 0

among the classes with Ξ ∈ κ′(g, r). Using test families, we will show that if Γ ∈ Tr(g) ∪T ′r(g)∪T ′′r (g)∪Lr−1,g0(g), then the coefficient cΓ in the linear relation has to be zero provingthat the class δΓ is independent from the classes indexed by κ′(g, r).

First, assume that Γ ∈ Tr(g)∪T ′r(g). Fix a general curve in ∆Γ. Recall that the curve has rcomponents C1, . . . , Cr attached to a central component C0. Let X be the r-dimensional testfamily obtained by varying the attachment point on the curves Ci for 1 ≤ i ≤ r. Every curvein the family X has the same dual graph Γ. Hence, [X] · δΞ = 0 for every Γ 6= Ξ ∈ κ′(g, r).By [E, Lemma 3.5], the intersection number [X] · δΓ =

∏ri=1(2 − 2g(Ci)). Since g(Ci) > 1,

[X] · δΓ 6= 0. Intersecting Relation (11), we conclude that cΓ = 0. Hence, δΓ is independentof the classes δΞ with Γ 6= Ξ ∈ κ′(g, r) and by Lemma 5.5 is an extremal codimension r class.

Next, suppose Γ ∈ Lr−1,g0(g). A general point of ∆Γ has r−1 curves C1, . . . , Cr−1 attachedto a one-nodal curve C0 of geometric genus g0. Let P be a general pencil of curves of type(2, g0+2) on P1×P1. Such a pencil has 4g0+8 base points. Let Y be the r-dimensional familyobtained by varying C0 in the pencil P and attaching C1, . . . , Cr−1 at distinct base-points ofP along varying points on C1, . . . , Cr−1. We need the inequality 4g0 +9 ≥ r to ensure that wecan form this family. The family Y intersects the codimension r boundary components exactlywhen a member of the pencil becomes nodal. Hence, Y intersects the codimension r boundarystratum ∆Γ and is disjoint from all other codimension r bounday strata. We conclude that[Y ] · δΞ = 0 for Γ 6= Ξ ∈ κ′(g, r). By [E, Lemma 3.4], [Y ] · δΓ = (8g0 + 12)

∏r−1i=1 (1− 2g(Ci)),


which is not zero since g(Ci) ≥ 2 for 0 ≤ i ≤ r − 1. Intersecting Relation (11) with Y , weconclude that cΓ = 0. Hence, δΓ is independent of the classes δΞ with Γ 6= Ξ ∈ κ′(g, r) andby Lemma 5.5 is an extremal codimension r class.

Finally, suppose that Γ ∈ T ′′r (g). A general point of ∆Γ parameterizes a curve with rcomponents C1, . . . , Cr attached to a central component C0, where 4g(C0) + 9 ≥ r. Inaddition, one of the components, say C1, has genus one. Let Q be a general pencil of planecubics. Let Z be the r-dimensional family obtained by attaching C0 to the pencil Q at abase point and varying the points of attachments on C2, . . . , Cr. The pencil Q has 12 nodalmembers. Since the geometric genus of the nodal curves in Q are zero, the only boundarystrata δΞ, with Ξ ∈ κ′(g, r), that Z intersects are ∆Γ and ∆Ψ with Ψ ∈ Lr−1,g(C0)(g). By [E,

Lemma 3.5], [Z]·δΓ = (−1)·∏ri=2(2−2g(Ci)) 6= 0. By the previous paragraph, the coefficients

cΨ in the Relation (11) are zero. Hence, intersecting the relation with Z, we conclude thatcΓ = 0. Hence, δΓ is independent of the classes δΞ with Γ 6= Ξ ∈ κ′(g, r) and by Lemma 5.5is an extremal codimension r class. This concludes the proof of the theorem. �

Remark 5.7. If 2r + 2 ≤ g, then the sets Tr(g), T ′r(g) and T ′′r (g) are non-empty. Hence,Theorem 5.6 gives examples of extremal cycles of arbitrarily high codimension. Note thatthe codimenions of these examples are ≤ g−2

2 , which is roughly one-sixth of the dimension

of Mg. It would be interesting to construct extremal cycles of arbitrarily high codimension

relative to the dimension of Mg.

Remark 5.8. By Remark 2.7, we further conclude that δΓ is extremal in the pseudoeffectivecone.

As mentioned before, it would be interesting to find an extremal higher codimension cyclenot contained in the boundary of Mg. Since for g ≥ 1 any birational morphism from Mg,n

has its exceptional locus contained in the boundary ([GKM, Corollary 0.11]), one cannotdirectly apply Proposition 2.1. Nevertheless, we are able to find a non-boundary extremalcodimension two cycle in M4.

Let H ⊂ M4 be the closure of the locus of hyperelliptic curves and let GP ⊂ M4 bethe closure of the Gieseker-Petri special curves, i.e. curves whose canonical embeddings arecontained in a quadric cone in P3.

Theorem 5.9. The cycle class of H is extremal in Eff2(M4).

Proof. The proof relies on analyzing the canonical model of a genus four curve C. If C is 3-connected and non-hyperelliptic, then the canonical embedding of C is a complete intersectionof a quadric surface Q and a cubic surface T in P3. If Q is smooth, consider the linear systemV = |O(3, 3)| on Q. Let U ⊂ V be the open locus of smooth curves. Let NI ⊂ V be thecodimension one locus of nodal irreducible curves. Let CP ⊂ V be the codimension two locusof irreducible curves with a cusp. Note that V \(U ∪NI ∪ CP ) has codimension three in V .Curves in U ∪NI are stable and curves in CP are pseudostable. In addition, NI dominates∆0 under the moduli map.

If Q is a quadric cone with vertex v, let F2 be the Hirzebruch surface obtained by blowingup v. Let E be the exceptional (−2)-curve with class e and let f be the ruling class of F2. If[C] ∈ GP is general, then C has class 3e + 6f as a curve in F2. Consider the linear systemV ′ = |3e + 6f | on F2. Let U ′ ⊂ V ′ be the open locus of smooth curves. Let NI ′1 ⊂ V ′ bethe codimension one locus of irreducible nodal curves. Let NI ′2 ⊂ V ′ be the codimensionone locus of irreducible at-worst-nodal curves B of class 2e + 6f union E, where B and E


intersect transversally at two distinct points. Let NI ′3 ⊂ V ′ be the codimension two locusof irreducible at-worst-nodal curves B of class 2e+ 6f union E, where B and E are tangentat one point. Let CP ′ ⊂ V ′ be the codimension two locus of irreducible curves with a cusp.Note that V ′\(U ′∪NI ′1∪NI ′2∪NI ′3∪CP ′) has codimension three in V ′. Curves in U ′∪NI ′1are stable and curves in CP ′ are pseudostable. For a curve in NI ′2, its stabilization as acurve in Q has a node at v. For a curve in NI ′3, its pseudostabilization as a curve in Q hasa cusp at v. In other words, blowing down E, curves in NI ′2 and in NI ′3 become stable andpseudostable, respectively.

Let h be the class of H. Suppose that

h =∑i

ai[Zi] ∈ N2(M4)(12)

for ai > 0 and Zi ⊂ M4 irreducible subvariety of codimension two, not equal to H. Recallthe first divisorial contraction ps :M4 →M

ps4 induced by replacing an elliptic tail by a cusp

for curves in ∆1. Applying ps∗, we obtain that

ps∗ h =∑i

ai(ps∗[Zi]) ∈ N2(Mps4 ).(13)

We first show that Zi is contained in GP ∪ ∆ for all i. By rearranging the indices ifnecessary, we concentrate on Z1. If ps∗[Z1] = 0, then Z1 ⊂ ∆1. Hence, we may assume thatps∗[Z1] 6= 0 in (13). Suppose that Z1 is not contained in GP ∪ ∆. Then a general curveC parameterized by Z1 is smooth and its canonical embedding is contained in a smoothquadric surface. Take a two-dimensional subspace S in V such that S is spanned by Cand two other general (3, 3)-curves. Then S is disjoint from V \(U ∪ NI ∪ CP ). Hence,any curve parameterized in S is pseudostable. Furthermore, since all these curves lie on asmooth quadric surface, the curves parameterized by S are not contained in ps(GP ) andnone of the curves are hyperelliptic. For a curve D parameterized in ps(Zi), if D does notadmit a canonical embedding contained in a smooth quadric, then [D] 6∈ S. On the otherhand if a general curve parameterized by ps(Zi) admits a canonical embedding contained ina smooth quadric, then ps(Zi) corresponds to a locus of codimension ≥ 2 in V . Therefore,[S] · (ps∗ h) = 0, [S] · (ps∗[Z1]) > 0 and [S] · (ps∗[Zi]) ≥ 0 for i 6= 1. Plugging in (13) leads toa contradiction. We thus conclude that Zi ⊂ GP ∪∆ for all i.

Next, we show that if Zi 6= H, then it is contained in ∆. Again we concentrate on Z1.Suppose that a general curve [C] ∈ Z1 6= H is contained in GP but not in ∆. In particular, Cis smooth and non-hyperelliptic. Consequently, its canonical model is a complete intersectionof a singular quadric Q and a cubic T . Let B1 and B2 be a pencil of quadrics and a pencilof cubics containing Q and T , respectively, and otherwise general. Let S = B1 × B2 be thetwo-dimensional family of canonical curves of genus four arising from the intersection of eachpair of these quadrics and cubics. Based on the previous analysis of V and V ′, we can assumethat all curves parameterized by S are pseudostable and none of them are hyperelliptic. Notethat B1 contains finitely many singular quadrics (including Q), hence S contains finitelymany one-dimensional families of curves that are parameterized in GP . Moreover, theseone-dimensional families are moving curves in GP by the construction of S. Similarly, theintersection of S with a boundary divisor consists of moving curves in the boundary. Hence,we can assume that S intersects each of the ps(Zi) in at most finitely many points. It followsthat [S] · (ps∗ h) = 0, [S] · (ps∗[Z1]) > 0 and [S] · (ps∗[Zi]) ≥ 0 for i 6= 1. Plugging in (13)leads to a contradiction. We thus conclude that if Zi 6= H, then Zi is contained in ∆.


Now it follows from (12) that h = 0 ∈ A2(M4), which contradicts the fact that h is anonzero multiple of λ2 in M4 ([Fab2]). �

6. The effective cones of M0,n

In this section, we study effective cycles on M0,n. Since dimM0,n = n − 3, there are nointeresting higher codimension cycles for n ≤ 6. Hence, from now on we assume that n ≥ 7.

Let S be a subset of {1, . . . , n} such that |S|, |Sc| ≥ 2. To make the notation symmetric,denote by ∆S,Sc the boundary divisor of M0,n parameterizing genus zero curves that havea node that separates the curve into two curves, one marked by S and the other by Sc.By definition, ∆S,Sc = ∆Sc,S . The codimension one boundary strata of M0,n consist of thedivisors ∆S,Sc , where the pair {S, Sc} varies over subsets of {1, . . . , n} with |S|, |Sc| ≥ 2.

Consider the codimension two boundary strata of M0,n. Let S1, S2, S3 be an ordereddecomposition of {1, . . . , n} with si = |Si| such that s1 + s2 + s3 = n, s1, s3 ≥ 2 ands2 ≥ 1. Let DS1,S2,S3 be the codimension two boundary stratum ofM0,n whose general pointparameterizes a chain of three smooth rational curves C1, C2, C3 such that Ci is marked bySi. By definition, DS1,S2,S3 = DS3,S2,S1 .

Theorem 6.1. The cycle class of DS1,S2,S3 is extremal in Eff2(M0,n).

Proof. By Proposition 2.1, it suffices to exhibit morphisms f such that ef (DS1,S2,S3) > ef (Z)for any subvariety Z of codimension two that is not DS1,S2,S3 . We will use the morphisms

fA :M0,n →M0,A introduced in Section 3.

First, suppose s1, s3 > 2. Let A be the weight parameter assigning 1s1

to the marked points

in S1 and 1 to the other marked points. Then efA(DS1,S2,S3) = s1−2 > 0. By Proposition 2.1,

if efA(Z) ≥ s1 − 2, then Z has to be contained in ∆S1,Sc1. Similarly, let B assign 1

s3to the

marked points in S3 and 1 to the other marked points. Then efB(DS1,S2,S3) = s3 − 2 > 0. IfefB(Z) ≥ s3 − 2, then Z is contained in ∆S3,Sc

3. Since the intersection of ∆S1,Sc

1and ∆S3,Sc

3

is exactly DS1,S2,S3 , we conclude that DS1,S2,S3 is extremal when s1, s3 > 2.

Next, suppose that s1 = s3 = 2. Since n ≥ 7, we have s2 = n − 4 ≥ 3. Without loss ofgenerality, let S1 = {1, 2}, S3 = {3, 4} and S2 = {5, . . . , n}. Let C assign 1

n−2 to the marked

points in S1∪S2 and 1 to the marked points in S3. We have efC(DS1,S2,S3) = s1+s2−3 = n−5.If efC(Z) ≥ n− 5, then Z is contained in ∆S,Sc , where S ⊂ {1, 2, 5, . . . , n} and |S| = n− 2 or

n− 3. Similarly let D assign 1n−2 to the marked points in S3 ∪S2 and 1 to the marked points

in S1. We have efD(DS1,S2,S3) = s2 + s3 − 3 = n− 5. If efD(Z) ≥ n− 5, then Z is containedin ∆T,T c where T ⊂ {3, 4, 5, . . . , n} and |T | = n − 2 or n − 3. When |S| = |T | = n − 2, theintersection of ∆S,Sc and ∆T,T c is exactly DS1,S2,S3 . If |S| = n− 2 and |T | = n− 3, then theintersection of ∆S,Sc and ∆T,T c is of the type DS′1,S

′2,S′3

where S′1 = {3, 4} and S′3 = {1, 2, i}for i ∈ S2. We have efD(DS′1,S

′2,S′3) = n−6 < efD(DS1,S2,S3). Finally, if |S| = |T | = n−3, the

intersection of ∆S,Sc and ∆T,T c is of the type DS′1,S′2,S′3

where S′1 = {3, 4, i} and S′3 = {1, 2, j}for i 6= j ∈ S2. We have efD(DS′1,S

′2,S′3) = n− 6 < efD(DS1,S2,S3). This proves that DS1,S2,S3

is extremal when s1 = s3 = 2.

The case s1 = 2 and s3 > 2 can be verified by a similar (and simpler) argument as in theabove paragraph, so we omit the details. �

Next, we give explicit examples of extremal higher codimension cycles in M0,n. LetS1, . . . , Sk be an unordered decomposition of {1, . . . , n} such that k ≥ 3 and |Si| ≥ 2 for


each i. Denote by BS1,...,Skthe subvariety of M0,n whose general point parameterizes a ra-

tional curve R attached to k rational tails C1, . . . , Ck such that Ci is marked by Si. Thecodimension of BS1,...,Sk

in M0,n equals k.

Theorem 6.2. The cycle class of BS1,...,Skis extremal in Effk(M0,n).

Proof. Suppose that [BS1,...,Sk] =

∑rj=1 aj [Zj ] ∈ Nk(M0,n), where aj > 0 and Zj ⊂ M0,n

is an irreducible subvariety of codimension k. First, consider the case |Si| = si > 2 for alli. Let Ai assign 1

sito the marked points in Si and 1 to the other marked points. Then

efAi(BS1,...,Sk

) = si− 2 > 0. If efAi(Z) ≥ si− 2, then Z has to be contained in ∆Si,Sc

i. Hence

by Proposition 2.1, Zj is contained in ∆S1,Sc1∩ · · · ∩∆Sk,S

ck

for all j. The intersection locusis exactly BS1,...,Sk

, which is irreducible. We thus conclude that Zj = BS1,...,Sk.

Next, consider the case k ≥ 4 and some si = 2. Without loss of generality, assume thatSi = {2i − 1, 2i} for 1 ≤ i ≤ l and sl+1, . . . , sk > 2. Let B0,n be the moduli space of n-pointed genus zero curves with rational k-fold singularities (without unmarked components).A rational k-fold singularity is locally isomorphic to the intersection of the k concurrentcoordinate axes in Ak. There is a birational morphism f :M0,n → B0,n contracting unmarked

components X to a rational k-fold singularity, where k = #(X ∩ C\X) ≥ 3, see e.g. [CC1,3.7–3.11]. Note that ef (BS1,...,Sk

) = k − 3 > 0. By Proposition 2.1, we have ef (Zj) ≥ k − 3.Since an unmarked component with m nodes loses (m− 3)-dimensional moduli under f , weconclude that Zj has to be one of the BT1,...,Tk . Using the morphism fAi in the previousparagraph for i > l, we further conclude that Zj is of the type BT1,...,Tl,Sl+1,...,Sk

, whereT1, . . . , Tl yield a decomposition of {1, . . . , 2l}. Since |Ti| ≥ 2, it implies that |Ti| = 2 foreach i. Therefore, Zj is the image of BS1,...,Sk

under the automorphism of M0,n induced byrelabeling {2i− 1, 2i} as Ti for each i. Note that the numerical classes of cycles in the orbitof BT1,...,Tl,Sl+1,...,Sk

by relabeling the marked points are not all proportional, which can beeasily checked using test families. Therefore, by symmetry of the marked points, any one ofthem cannot be a nonnegative linear combination of the others.

The remaining case is that k = 3 and some si = 2. Since n ≥ 7, without loss of generality,assume that S1 = {1, 2}, S2 = {3, . . . ,m} and S3 = {m+1, . . . , n} with 4 ≤ m ≤ n−3 so thats3 = n−m > 2. As above, we know that Zj is contained in ∆S3,Sc

3. By [K, p. 549], we know

that N2(∆S3,Sc3)→ N3(M0,n) is injective. Moreover, for any curve parameterized in ∆S3,Sc

3,

there exists a unique node that separates the curve into two connected components, containingthe marked points in S3 and in Sc3, respectively. It follows that ∆S3,Sc

3∼=M0,m+1×M0,n−m+1

(see also [K, p. 548]). Similarly BS1,S2,S3 can be identified with DS1,{p},S2×M0,n−m+1 ⊂

∆S3,Sc3∼=M0,m+1×M0,n−m+1, where p denotes the unique node that separates the curve into

two components, containing the marked points in S3 and in Sc3, respectively. By Theorem 6.1,DS1,{p},S2

is an extremal codimension two cycle in M0,m+1, hence BS1,S2,S3 is an extremalcodimension two cycle in ∆S3,Sc

3by Corollary 2.4. It follows that BS1,S2,S3 is an extremal

codimension three cycle in M0,n by Proposition 2.5. �

Finally, we point out that not all boundary strata are extremal cycles on moduli spaces ofpointed stable curves.

Remark 6.3. Recall that M0,n =M0,n/Sn is the moduli space of stable genus zero curves

with n unordered marked points. The one-dimensional strata of M0,n (F -curves) correspond

to partitions of n into four parts. For n = 7, there are three F -curves on M0,7: F1,1,1,4, F1,1,2,3


and F1,2,2,2. However, their numerical classes satisfy that 2[F1,1,2,3] = [F1,1,1,4] + [F1,2,2,2] (see

e.g. [M, Table 1]). In particular, F1,1,2,3 is not extremal in the Mori cone of curves on M0,7.


In this section, using the infinitely many extremal effective divisors inM1,n−2 constructed

in [CC2], we show that Eff2(M1,n) is not finite polyhedral for n ≥ 5.

Let p1, . . . , pn be the n marked points and let T = {pn−2, pn−1, pn}. The gluing morphism

∆0;T =M1,n−2 ×M0,4 → ∆0;T ⊂M1,n

is induced by gluing a pointed genus one curve (E, p1, . . . , pn−3, q) to a pointed rational curve(C, pn−2, pn−1, pn, q) by identifying q in E and C to form a node. Denote by ΓS (resp. Γ0)the image of ∆0;S ×M0,4 (resp. ∆0 ×M0,4) in M1,n for S ⊂ {p1, . . . , pn−3, q} and |S| ≥ 2.

Let Γ be the image of M1,n−2 ×∆0;{pn−1,pn}. Note that Γ0, ΓS and Γ are the codimension

two boundary strata of M1,n whose general point parameterizes a two-nodal curve that hasa rational tail marked by T . The cases q ∈ S and q 6∈ S correspond to the middle componenthaving genus zero and one, respectively.

Since M0,4∼= P1, it follows that A1(∆0;T ) ∼= N1(∆0;T ), generated by the classes of ΓS , Γ0

and Γ.

Lemma 7.1. The cycle classes of ΓS, Γ0 and Γ are independent in N2(M1,n).

Proof. Denote by γS , γ0 and γ the classes of ΓS , Γ0 and Γ, respectively. Suppose that theysatisfy a relation

(14) a · γ0 +∑S

bS · γS + c · γ = 0

in N2(M1,n). Below we will show that all the coefficients in (14) are zero.

Let A be the weight parameter assigning 13 to pn−2, pn−1, pn and 1 to pk for k ≤ n − 3.

Then efA(Γ) = 0 and efA(Γ0) = efA(ΓS) = 1. Applying fA∗ to (14), we conclude that c = 0.Relation (14) reduces to

(15) a · γ0 +∑S

bS · γS = 0.

Let ψ :M1,n →M1,n−2 be the morphism forgetting p1 and p2. We have eψ(Γ{p1,p2}) = 0and eψ(Γ0), eψ(ΓS) > 0 for S 6= {p1, p2}. Applying ψ∗ to (15), we conclude that b{p1,p2} = 0,

hence by symmetry bS = 0 for S = {pi, pj} for i, j ≤ n − 3. Next, let φ : M1,n → M1,n−1

be the morphism forgetting p1. Among the remaining cycles, eφ(Γ{p1,q}) = 0 and eφ(Γ0) =eφ(ΓS) > 0 for S 6= {p1, q}. Applying φ∗, we conclude that b{p1,q} = 0, hence by symmetryb{pi,q} = 0 for all i ≤ n− 3. In sum, we obtain that bS = 0 for all |S| = 2.

Apply induction on |S|. Suppose that bS = 0 for all |S| ≤ k − 1. Relation (15) reduces to

(16) a · γ0 +∑|S|≥k

bS · γS = 0.

Let ϕ :M1,n →M1,n−k be the morphism forgetting p1, . . . , pk. Then we have eϕ(Γ{p1,...,pk}) =k − 2 and eϕ(Γ0), eϕ(ΓS) > k − 2 for |S| ≥ k and S 6= {p1, . . . , pk}. Take an ample divisor

class A inM1,n, intersect (16) with Ak−2 and apply ϕ∗. We obtain that b{p1,...,pk} = 0, hence


by symmetry bS = 0 for S ⊂ {p1, . . . , pn−3} and |S| = k. Next, let η :M1,n →M1,n−k+1 bethe morphism forgetting p1, . . . , pk−1. Among the remaining cycles, eη(Γ{p1,...,pk−1,q}) = k−2

and eη(Γ0), eη(ΓS) ≥ k−1 for S 6= {p1, . . . , pk−1, q}. Intersecting with Ak−2 and applying η∗,we obtain that b{p1,...,pk−1,q} = 0, hence bS = 0 for all |S| = k. By induction we thus concludethat bS = 0 for all S.

Finally, Relation (16) reduces to a · γ0 = 0, hence a = 0. �

Theorem 7.2. For n ≥ 5, Eff2(M1,n) is not finite polyhedral.

Proof. Let A be the weight parameter assigning 13 to marked points in T = {pn−2, pn−1, pn}

and 1 to the other marked points. The exceptional locus of fA :M1,n →M1,A is ∆0;T . By

Corollary 2.4, any extremal effective divisor D′ on M1,n−2 pulls back to an extremal divisor

D on ∆0;T =M1,n−2 ×M0,4. Moreover, efA(D) > 0 because the moduli of the rational tailmarked by T is forgotten under fA. By Lemma 7.1, we can apply Proposition 2.5 to concludethat the class of D is extremal in Eff2(M1,n). Now the claim follows from the fact that there

are infinitely many extremal effective divisors on M1,n−2 for every n ≥ 5 ([CC2]). �


Applying the same idea as in Section 7, in this section we show that Eff2(M2,n) is notfinite polyhedral for n ≥ 2.

Let p1, . . . , pn be the n marked points. The gluing morphism

∆1;∅ =M1,n+1 ×M1,1 → ∆1;∅ ⊂M2,n

is induced by gluing two pointed genus one curves (E1, p1, . . . , pn, q) and (E2, q) by identifyingq in both curves to form a node. Denote by ΓS (resp. Γ0) the image of ∆0;S ×M1,1 (resp.

∆0×M1,1) inM2,n for S ⊂ {1, . . . , n+ 1} and |S| ≥ 2. Let Γ be the image ofM1,n+1×∆0.

Note that Γ0, ΓS and Γ are the codimension two boundary strata of M2,n whose generalpoint parameterizes a curve with two nodes and an unmarked tail of arithmetic genus one.The cases q ∈ S and q 6∈ S correspond to the middle component having genus zero and one,respectively.

Since M1,1∼= P1, it follows that A1(∆1;∅) ∼= N1(∆1;∅), generated by the classes of ΓS , Γ0

and Γ.

Lemma 8.1. The cycle classes of ΓS, Γ0 and Γ are independent in N2(M2,n).

Proof. Denote by γS , γ0 and γ the classes of ΓS , Γ0 and Γ in M2,n, respectively. Supposethat they satisfy

(17) a · γ0 +∑S

bS · γS + c · γ = 0.

Recall that ps :M2,n →Mps2,n contracts ∆1;∅, replacing an elliptic tail by a cusp. Hence we

have eps(Γ) = 0 and eps(Γ0) = eps(ΓS) = 1 for all S. Applying ps∗ to (17), we conclude thatc = 0. Now the same induction procedure in the proof of Lemma 7.1 implies that a = bS = 0for all S. �

Theorem 8.2. For n ≥ 2, Eff2(M2,n) is not finite polyhedral.


Proof. The exceptional locus of ps : M2,n → Mps2,n is ∆1;∅. By Corollary 2.4, any extremal

effective divisor D′ on M1,n+1 pulls back to an extremal divisor D on ∆1;∅ = M1,n+1 ×M1,1. Moreover, eps(D) > 0 because the unmarked genus one tail is forgotten under ps.By Lemma 8.1, we can apply Proposition 2.5 to conclude that the class of D is extremalin Eff2(M2,n). Now the claim follows from the fact that there are infinitely many extremal

effective divisors on M1,n+1 for every n ≥ 2 ([CC2]). �

Remark 8.3. Since N1(∆1) ∼= A1(∆1)→ N2(Mg) is injective and ∆1∼=Mg−1,1×M1,1, the

same proof as that of Theorem 8.2 implies that the pullback of any extremal effective divisor

fromMg−1,1 to ∆1 gives rise to an extremal codimension two cycle inMg. For instance, the

divisor W of Weierstrass points is known to be extremal in Mg−1,1 for 3 ≤ g ≤ 6 (see e.g.[C, Theorem 4.3 and Remark 4.4]). Hence W yields an extremal codimension two cycle inEff2(Mg) for 3 ≤ g ≤ 6.

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Department of Mathematics, Boston College, Chestnut Hill, MA 02467

E-mail address: [email protected]

Department of Mathematics, Statistics, and Computer Science, University of Illinois atChicago, Chicago, IL 60607

E-mail address: [email protected]

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